Note that one can calculate dφ by the area of sector confined by
these vectors dφ = (s1 s2' -
s1' s2 ) dx .

The Sx2
→ Sn1 maps

If at infinity n(x,y) → no ,
then 2D plane (x, y) is equivalent topologically to a sphere.
But one can shrink into a point any map of a sphere to a circle.
Therefore any smooth field is equivalent to the trivial field
n = no .
It is true for any D > 1 too.

Vorteces

Topoligical "deffects" on a plane appear if one take a contour with
Q ≠ 0 . Then there are vorteces within it. In three
dimensional space one shall draw a surface on this contour and then vortex
threads cross it.

The circle to sphere map

For the continuum XY-chain with Q ≠ 0 above it is impossible
to move upwords smoothly all "blue" arrows rotating spins in the picture plane.
But one makes it easy by rotation around the x axis in 3D space
(i.e. for the Heisenberg spins). Therefore Heisenberg chains have no
topological excitations.
It is equivalent to the statement that one can shrink into a point any map
of a circle to a sphere (i.e. any contour on a sphere).

But metastable states will appear in the Heisenberg model on 2D plane.